A solid cylinder rolls down an incline. If its height is h, what is its linear s

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Question: A solid cylinder rolls down an incline. If its height is h, what is its linear speed at the bottom? (2023)

Options:

Correct Answer: √(2gh)

Solution:

Using conservation of energy, potential energy converts to kinetic energy. For a solid cylinder, v = √(2gh).

A solid cylinder rolls down an incline. If its height is h, what is its linear speed at the bottom? (2023)

A solid cylinder rolls down an incline. If its height is h, what is its linear speed at the bottom? (2023)

Step 1: Understand that the solid cylinder starts at a height 'h' on the incline.
Step 2: Recognize that at the top, the cylinder has potential energy due to its height.
Step 3: Know that as the cylinder rolls down, this potential energy converts into kinetic energy.
Step 4: Remember the formula for potential energy: PE = mgh, where m is mass, g is gravity, and h is height.
Step 5: Understand that the kinetic energy (KE) of a rolling object includes both translational and rotational energy.
Step 6: For a solid cylinder, the total kinetic energy at the bottom can be expressed as KE = (1/2)mv^2 + (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity.
Step 7: For a solid cylinder, the moment of inertia I = (1/2)mr^2 and the relationship between linear speed (v) and angular speed (ω) is ω = v/r.
Step 8: Substitute I and ω into the kinetic energy equation to express it in terms of v.
Step 9: Set the potential energy equal to the total kinetic energy to find the relationship between height and speed.
Step 10: Solve the equation to find the linear speed v at the bottom of the incline: v = √(2gh).

Conservation of Energy – The principle that energy cannot be created or destroyed, only transformed from one form to another.
Kinetic Energy of Rolling Objects – For rolling objects, kinetic energy includes both translational and rotational components.
Potential Energy – The energy stored due to an object's height above a reference point, calculated as mgh.